Monotonicity methods for Mean-Field Games - Tada, Teruo (2021)
Mean-field games (MFGs) model the behavior of large populations of rational agents. Each agent seeks to minimize an individual cost that depends on the statistical distribution of the population. Roughly speaking, MFGs are given by the limit of differential games with N agents as N goes to infinity. This limit describes an average effect of the population’s behavior. Instead of modeling large systems for all agents, we consider two coupled equations: the Hamilton–Jacobi equation and the Fokker–Planck equation. A solution to MFGs is given by two functions: a value function and a population density. From the point of view of mathematics, monotonicity conditions for MFGs are a natural way to obtain the uniqueness of solutions and the stability of systems.